Consider an (L, α)-superdiffusion X on ℝd, where L is an uniformly elliptic differential operator in ℝd, and 1 < α < 2. The G-polar sets for X are subsets of ℝ × ℝd which have no intersection with the graph G of X, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the G-polarity of a general analytic set A ⊂ ℝ × ℝd in term of the Bessel capacity of A, and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the G-polarity of sets of the form E × F, where E and F are two Borel subsets of ℝ and ℝd respectively. We establish a relationship between the restricted Hausdorff dimension of E × F and the usual Hausdorff dimensions of E and F. As an application, we obtain a criterion for G-polarity of E × F in terms of the Hausdorff dimensions of E and F, which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.
|Number of pages||8|
|Journal||Proceedings of the American Mathematical Society|
|State||Published - 1 Dec 1999|
- Box dimension
- Graph of superdiffusion, semilinear partial differential equation
- Hausdorff dimension
- Restricted hausdorff dimension